In "Audie Murphy American Soldier", page 258 it is stated that Jimmy Cagney's residence was 2001 Cold Water Canyon in Beverly Hills. The book doesn't make it clear if Audie was actually living there at the time of this letter, but had lived there in late '45 and most of '46, according to the book.
Leon, you are right. I bet you my bottom dollar Audie had a nice quantity of the stationary printed before he moved out of the Cagney residence and couldn't bear to waste money on throwing it away even though his address was not the same. The address makes the letter even more "valuable" just because it corroborates one of the details of history we know. Great stuff!